NET::ERR_CERT_DATE_INVALID "Privacy Error Your Connection is not Private"

...will I just have to wait for those websites to update their security to access them again? Or will I never be able to access them.

Based of history, the chances of web developers and certificate issuers reverting to supporting older versions or Mac or Windows OS is slim to none. Note old versions of Windows are also affected. So "never: is likely the operative response.

What can I do apart from just using Firefox to access websites?

Other than a new computer or Firefox, nothing that I know. I would watch the "Let's encrypt" web site I linked above in the off-chance that they do something for older Win and Mac OS versions. But that is a loooong shot.

We replaced three Macs over the years solely because Intuit withdrew TurboTax support for older OS versions. So this is not very surprising to me.

More info in what zarathu has given:

Pretty much all the reports here of seeing that advisory are from posters running macOS 10.11 El Capitan or older.

A major supplier of browser security certificates has allowed some of its certs to expire. This primarily affects OS versions with outdated browsers. However, I found a few sites showing the message even with modern supported browsers (Safari 14).

Here is that supplier's statement on the subject:

https://letsencrypt.org/docs/dst-root-ca-x3-expiration-september-2021/

Older devices and/or their OS versions and/or browsers may not support technologies that the cert's replacement uses. To get a newer version of Safari usually requires upgrading to a newer macOS version, as you have found. And you cannot upgrade the model any farther.

BTW, I am getting that same advisory from WikiPedia when I use an old iPad running iOS 9.3.5, which is roughly contemporary with macOS El Capitan.

However, something to try first: If you are seeing that advisory in Safari or Chrome, try FireFox. I'm been told by trusted sources that FireFox handles certs differently from other browsers and sites with the expired cert may still work.

https://www.mozilla.org

However, even FireFox is giving up on El Capitan and will move you to their Extended Support Release (ESR) version which as of last July, is not as secure as the full version.

So it is either try FireFox ESR which, at least for now, still supports El Capitan, or shop for a new computer. In the long run, the handwriting is on the wall for the computer you now have.

The are other browsers but support for El Capitan is dropping like flies in an insecticide testing lab. You can search for mac browsers but very few support 10.11 any more.

Choose Utilities from the Finder's Go menu, open the Script Editor, and run:

do shell script "curl -k https://letsencrypt.org/certs/isrgrootx1.der > ~/Downloads/isrgrootx1.der"

Double-click the downloaded certificate and set Keychain Access to always trust it.

(207785)

This is a problem with the websites that you are going to, not a problem with your computer. It is simply telling you that that these websites have not kept up with their internet security, and that because of that if you go there, your mac is not sure if the site is a real and it may try to hack into your computer.

Usually it doesn’t happen. But then again, you are running a very very old mac, which has not had any security upgrades for the OS for a very long time. Security upgrades stop when the OS gets too obsolete.

Alright. There is a way. Stop using the Mac interface on your computer. Boot up Linux OS instead. Versions of it will run on the oldest macs and PC’s without issue and some of them are very current OS’s.

But doing that is not an easy thing to do. I spend nearly a year trying to get Linux Mint to work properly on a Dell from 2006, but was eventually able to make it happen.

We cannot help you do that here, unfortunately, but it is a way to access very old websites on very old computers running a new system.

"Very old old websites on very old computers running a new system"

What is wrong with macOS and why is Linux the only option?

This has been explained here over and over and over. Please take your ranting elsewhere.

Where has it been explained?